direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C22⋊Q8, C23⋊3(C5×Q8), C4.63(D4×C10), (C22×C10)⋊6Q8, C22⋊1(Q8×C10), (C2×C20).524D4, C20.470(C2×D4), (C22×Q8)⋊3C10, C24.31(C2×C10), (C23×C20).25C2, (C23×C4).10C10, (Q8×C10)⋊48C22, C22.60(D4×C10), C10.57(C22×Q8), (C2×C10).343C24, (C2×C20).656C23, C10.182(C22×D4), (C23×C10).91C22, C23.70(C22×C10), C22.17(C23×C10), (C22×C20).444C22, (C22×C10).258C23, C2.6(D4×C2×C10), C2.3(Q8×C2×C10), (C10×C4⋊C4)⋊42C2, (C2×C4⋊C4)⋊15C10, (Q8×C2×C10)⋊15C2, (C2×C10)⋊5(C2×Q8), C4⋊C4⋊10(C2×C10), (C2×Q8)⋊8(C2×C10), C2.6(C10×C4○D4), (C5×C4⋊C4)⋊66C22, (C2×C4).135(C5×D4), C10.225(C2×C4○D4), (C2×C10).682(C2×D4), C22.30(C5×C4○D4), (C10×C22⋊C4).31C2, (C2×C22⋊C4).11C10, C22⋊C4.10(C2×C10), (C2×C4).12(C22×C10), (C22×C4).53(C2×C10), (C2×C10).230(C4○D4), (C5×C22⋊C4).144C22, SmallGroup(320,1525)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C22⋊Q8
G = < a,b,c,d,e | a10=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 450 in 322 conjugacy classes, 194 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C5, C2×C4 [×16], C2×C4 [×18], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×8], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C2×Q8 [×4], C2×Q8 [×4], C24, C20 [×4], C20 [×10], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4 [×2], C2×C4⋊C4, C2×C4⋊C4 [×2], C22⋊Q8 [×8], C23×C4, C22×Q8, C2×C20 [×16], C2×C20 [×18], C5×Q8 [×8], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C5×C22⋊C4 [×8], C5×C4⋊C4 [×12], C22×C20 [×2], C22×C20 [×8], C22×C20 [×4], Q8×C10 [×4], Q8×C10 [×4], C23×C10, C10×C22⋊C4 [×2], C10×C4⋊C4, C10×C4⋊C4 [×2], C5×C22⋊Q8 [×8], C23×C20, Q8×C2×C10, C10×C22⋊Q8
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], Q8 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C2×C10 [×35], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, C5×D4 [×4], C5×Q8 [×4], C22×C10 [×15], C2×C22⋊Q8, D4×C10 [×6], Q8×C10 [×6], C5×C4○D4 [×2], C23×C10, C5×C22⋊Q8 [×4], D4×C2×C10, Q8×C2×C10, C10×C4○D4, C10×C22⋊Q8
►On 160 points
Generators in S160
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30)(31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50)(51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70)(71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90)(91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110)(111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130)(131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150)(151 152 153 154 155 156 157 158 159 160)
(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 28)(19 29)(20 30)(31 160)(32 151)(33 152)(34 153)(35 154)(36 155)(37 156)(38 157)(39 158)(40 159)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 107)(92 108)(93 109)(94 110)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)(111 125)(112 126)(113 127)(114 128)(115 129)(116 130)(117 121)(118 122)(119 123)(120 124)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(1 79 56 107)(2 80 57 108)(3 71 58 109)(4 72 59 110)(5 73 60 101)(6 74 51 102)(7 75 52 103)(8 76 53 104)(9 77 54 105)(10 78 55 106)(11 146 34 118)(12 147 35 119)(13 148 36 120)(14 149 37 111)(15 150 38 112)(16 141 39 113)(17 142 40 114)(18 143 31 115)(19 144 32 116)(20 145 33 117)(21 140 153 122)(22 131 154 123)(23 132 155 124)(24 133 156 125)(25 134 157 126)(26 135 158 127)(27 136 159 128)(28 137 160 129)(29 138 151 130)(30 139 152 121)(41 94 65 86)(42 95 66 87)(43 96 67 88)(44 97 68 89)(45 98 69 90)(46 99 70 81)(47 100 61 82)(48 91 62 83)(49 92 63 84)(50 93 64 85)
(1 119 56 147)(2 120 57 148)(3 111 58 149)(4 112 59 150)(5 113 60 141)(6 114 51 142)(7 115 52 143)(8 116 53 144)(9 117 54 145)(10 118 55 146)(11 78 34 106)(12 79 35 107)(13 80 36 108)(14 71 37 109)(15 72 38 110)(16 73 39 101)(17 74 40 102)(18 75 31 103)(19 76 32 104)(20 77 33 105)(21 82 153 100)(22 83 154 91)(23 84 155 92)(24 85 156 93)(25 86 157 94)(26 87 158 95)(27 88 159 96)(28 89 160 97)(29 90 151 98)(30 81 152 99)(41 134 65 126)(42 135 66 127)(43 136 67 128)(44 137 68 129)(45 138 69 130)(46 139 70 121)(47 140 61 122)(48 131 62 123)(49 132 63 124)(50 133 64 125)
G:=sub<Sym(160)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,146,34,118)(12,147,35,119)(13,148,36,120)(14,149,37,111)(15,150,38,112)(16,141,39,113)(17,142,40,114)(18,143,31,115)(19,144,32,116)(20,145,33,117)(21,140,153,122)(22,131,154,123)(23,132,155,124)(24,133,156,125)(25,134,157,126)(26,135,158,127)(27,136,159,128)(28,137,160,129)(29,138,151,130)(30,139,152,121)(41,94,65,86)(42,95,66,87)(43,96,67,88)(44,97,68,89)(45,98,69,90)(46,99,70,81)(47,100,61,82)(48,91,62,83)(49,92,63,84)(50,93,64,85), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,134,65,126)(42,135,66,127)(43,136,67,128)(44,137,68,129)(45,138,69,130)(46,139,70,121)(47,140,61,122)(48,131,62,123)(49,132,63,124)(50,133,64,125)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30)(31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50)(51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70)(71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90)(91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110)(111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130)(131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150)(151,152,153,154,155,156,157,158,159,160), (11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,62)(2,63)(3,64)(4,65)(5,66)(6,67)(7,68)(8,69)(9,70)(10,61)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,28)(19,29)(20,30)(31,160)(32,151)(33,152)(34,153)(35,154)(36,155)(37,156)(38,157)(39,158)(40,159)(41,59)(42,60)(43,51)(44,52)(45,53)(46,54)(47,55)(48,56)(49,57)(50,58)(71,85)(72,86)(73,87)(74,88)(75,89)(76,90)(77,81)(78,82)(79,83)(80,84)(91,107)(92,108)(93,109)(94,110)(95,101)(96,102)(97,103)(98,104)(99,105)(100,106)(111,125)(112,126)(113,127)(114,128)(115,129)(116,130)(117,121)(118,122)(119,123)(120,124)(131,147)(132,148)(133,149)(134,150)(135,141)(136,142)(137,143)(138,144)(139,145)(140,146), (1,79,56,107)(2,80,57,108)(3,71,58,109)(4,72,59,110)(5,73,60,101)(6,74,51,102)(7,75,52,103)(8,76,53,104)(9,77,54,105)(10,78,55,106)(11,146,34,118)(12,147,35,119)(13,148,36,120)(14,149,37,111)(15,150,38,112)(16,141,39,113)(17,142,40,114)(18,143,31,115)(19,144,32,116)(20,145,33,117)(21,140,153,122)(22,131,154,123)(23,132,155,124)(24,133,156,125)(25,134,157,126)(26,135,158,127)(27,136,159,128)(28,137,160,129)(29,138,151,130)(30,139,152,121)(41,94,65,86)(42,95,66,87)(43,96,67,88)(44,97,68,89)(45,98,69,90)(46,99,70,81)(47,100,61,82)(48,91,62,83)(49,92,63,84)(50,93,64,85), (1,119,56,147)(2,120,57,148)(3,111,58,149)(4,112,59,150)(5,113,60,141)(6,114,51,142)(7,115,52,143)(8,116,53,144)(9,117,54,145)(10,118,55,146)(11,78,34,106)(12,79,35,107)(13,80,36,108)(14,71,37,109)(15,72,38,110)(16,73,39,101)(17,74,40,102)(18,75,31,103)(19,76,32,104)(20,77,33,105)(21,82,153,100)(22,83,154,91)(23,84,155,92)(24,85,156,93)(25,86,157,94)(26,87,158,95)(27,88,159,96)(28,89,160,97)(29,90,151,98)(30,81,152,99)(41,134,65,126)(42,135,66,127)(43,136,67,128)(44,137,68,129)(45,138,69,130)(46,139,70,121)(47,140,61,122)(48,131,62,123)(49,132,63,124)(50,133,64,125) );
G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30),(31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50),(51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70),(71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90),(91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110),(111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130),(131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150),(151,152,153,154,155,156,157,158,159,160)], [(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,62),(2,63),(3,64),(4,65),(5,66),(6,67),(7,68),(8,69),(9,70),(10,61),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,28),(19,29),(20,30),(31,160),(32,151),(33,152),(34,153),(35,154),(36,155),(37,156),(38,157),(39,158),(40,159),(41,59),(42,60),(43,51),(44,52),(45,53),(46,54),(47,55),(48,56),(49,57),(50,58),(71,85),(72,86),(73,87),(74,88),(75,89),(76,90),(77,81),(78,82),(79,83),(80,84),(91,107),(92,108),(93,109),(94,110),(95,101),(96,102),(97,103),(98,104),(99,105),(100,106),(111,125),(112,126),(113,127),(114,128),(115,129),(116,130),(117,121),(118,122),(119,123),(120,124),(131,147),(132,148),(133,149),(134,150),(135,141),(136,142),(137,143),(138,144),(139,145),(140,146)], [(1,79,56,107),(2,80,57,108),(3,71,58,109),(4,72,59,110),(5,73,60,101),(6,74,51,102),(7,75,52,103),(8,76,53,104),(9,77,54,105),(10,78,55,106),(11,146,34,118),(12,147,35,119),(13,148,36,120),(14,149,37,111),(15,150,38,112),(16,141,39,113),(17,142,40,114),(18,143,31,115),(19,144,32,116),(20,145,33,117),(21,140,153,122),(22,131,154,123),(23,132,155,124),(24,133,156,125),(25,134,157,126),(26,135,158,127),(27,136,159,128),(28,137,160,129),(29,138,151,130),(30,139,152,121),(41,94,65,86),(42,95,66,87),(43,96,67,88),(44,97,68,89),(45,98,69,90),(46,99,70,81),(47,100,61,82),(48,91,62,83),(49,92,63,84),(50,93,64,85)], [(1,119,56,147),(2,120,57,148),(3,111,58,149),(4,112,59,150),(5,113,60,141),(6,114,51,142),(7,115,52,143),(8,116,53,144),(9,117,54,145),(10,118,55,146),(11,78,34,106),(12,79,35,107),(13,80,36,108),(14,71,37,109),(15,72,38,110),(16,73,39,101),(17,74,40,102),(18,75,31,103),(19,76,32,104),(20,77,33,105),(21,82,153,100),(22,83,154,91),(23,84,155,92),(24,85,156,93),(25,86,157,94),(26,87,158,95),(27,88,159,96),(28,89,160,97),(29,90,151,98),(30,81,152,99),(41,134,65,126),(42,135,66,127),(43,136,67,128),(44,137,68,129),(45,138,69,130),(46,139,70,121),(47,140,61,122),(48,131,62,123),(49,132,63,124),(50,133,64,125)])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 10A | ··· | 10AB | 10AC | ··· | 10AR | 20A | ··· | 20AF | 20AG | ··· | 20BL |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | C10 | C10 | D4 | Q8 | C4○D4 | C5×D4 | C5×Q8 | C5×C4○D4 |
| kernel | C10×C22⋊Q8 | C10×C22⋊C4 | C10×C4⋊C4 | C5×C22⋊Q8 | C23×C20 | Q8×C2×C10 | C2×C22⋊Q8 | C2×C22⋊C4 | C2×C4⋊C4 | C22⋊Q8 | C23×C4 | C22×Q8 | C2×C20 | C22×C10 | C2×C10 | C2×C4 | C23 | C22 |
| # reps | 1 | 2 | 3 | 8 | 1 | 1 | 4 | 8 | 12 | 32 | 4 | 4 | 4 | 4 | 4 | 16 | 16 | 16 |
Matrix representation of C10×C22⋊Q8 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 25 | 0 |
| 0 | 0 | 0 | 0 | 25 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 40 | 39 |
| 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 14 |
| 0 | 0 | 0 | 14 | 7 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,18,0,0,0,0,0,18,0,0,0,0,0,25,0,0,0,0,0,25],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,1,0,0,0,39,1],[1,0,0,0,0,0,0,40,0,0,0,40,0,0,0,0,0,0,34,14,0,0,0,14,7] >;
C10×C22⋊Q8 in GAP, Magma, Sage, TeX
C_{10}\times C_2^2\rtimes Q_8 % in TeX
G:=Group("C10xC2^2:Q8"); // GroupNames label
G:=SmallGroup(320,1525);
// by ID
G=gap.SmallGroup(320,1525);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,1149,568,3446]);
// Polycyclic
G:=Group<a,b,c,d,e|a^10=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations